Quantum integral Berwald and related inequalities

We establish quantum-integral analogues of Berwald-type inequalities on finite intervals. Starting from a generalized q -Favard inequality for nonnegative, nondecreasing functions whose \(1/k\) -th power is concave, we derive a q -Berwald inequality and weighted and product-type extensions. The arguments use integral comparison principles developed in earlier work on quantum Favard inequalities. Three explicit power-function examples illustrate the estimates and their dependence on the quantum parameter \(q∈ (0,1)\) . We also identify the classical limits as \(q→ 1⁻\) . The scope of the results is determined by the stated concavity and monotonicity hypotheses; no corresponding general assertion is made for arbitrary convex functions.